1. LINEAR KINEMATICS
KIN3323: Biomechanics

2. TYPES OF MOTION

3. Kinematics
Description of the form, pattern, or sequencing of
movement with respect to time
– Involves: position, velocity and acceleration of a body
without concern for the forces which cause motion
– Does not involve force

4. Types of Motion
1. Linear Motion
– Translation: all points on an object move the same
distance, in the same direction, and at the same time’
– 2 Types
a) Rectilinear: Movement in straight line path
b) Curvilinear: Movement in straight curved path

5. 1a. Rectilinear Translation (Linear Motion)
Example
• Figure skater gliding across the ice in a static position

6. 1a. Rectilinear Translation (Linear Motion)
Example
• Figure skater gliding across the ice in a static position

7. 1b. Curvilinear Translation (Linear Motion)
Example
• Skateboarder in air holding a static position
• Ski jumper

8. Types of Motion
2. Angular Motion
– Rotation: all points on an object moves in circles
about the same fixed axis

9. 2. Angular Motion
Example
• Elbow flexion
• Figure skater spinning

10. Types of Motion
3. General Motion
– Combination of translation and rotation
– Most common type of motion in sports and human
movement

11. DESCRIPTION OF
MOTION

12. Description of motion
• Position
• Distance / displacement
• Speed / velocity
• Acceleration

13. Description of motion: Position
• The location of a point, with respect to the origin,
within a spatial reference frame
• Reference frames
– Cartesian coordinate system
• Fixed point (origin) with axes that are perpendicular
to each other
• 2-D or 3-D system

14. 2-D Coordinate System
Used when motion is +y
primarily in one plane
• x (horizontal)
• y (vertical)
-x +x
Origin = (0,0)
(0,0) means that the point
is located at x=0 & y=0
ex. (2,5) describes the
-y point located at x=2 & y=5

15. Example of reference frames in Biomechanics
+y +y
-x +x -x +x
Origin Origin
-y -y
Global reference frame Anatomical reference frame
Relative to gravity Relative to body segment
(ex. forearm relative to arm)

16. Distance vs. Displacement
Distance Displacement
• Length of path • Straight line distance
followed by object in a specific direction
from initial to final from initial to final
position position
• Scalar quantity • Vector quantity
– Only magnitude – Has magnitude and
direction

17. Distance = length of path
y Distance
=
1+1.5+3+4+1+2+3.8+2
=
17.8
12
11
1.5m
10
1m
3m
2m
9
(11,10)
3.8m
8
7
(2,7)
6
5
4m
2m
4
3
1m
2
1
x
(0,0) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

18. Displacement = Straight line distance
y
12
11
10
9
(11,10)
Δy
8
7
(2,7) Δx
6
5
4
3
2
1
x
(0,0) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

19. Displacement = Straight line distance
(11,10)
Δy = 10 – 7 = 3
(2,7)
Δx = 11 – 2 = 9
Displacement2 = Δx2 + Δy2
Displacement = Δx2 + Δy2 = 92 + 32
= 9.5

20. Which is more relevant: Distance or displacement?
400m
Race

21. Which is more relevant: Distance or displacement?
400m
Race
Displacement?
Distance?
Start
End

22. Which is more relevant: Distance or displacement?
Javelin
throw

23. Which is more relevant: Distance or displacement?
Distance
=
200m
Distance
=
200m
Displacement
Displacement
=
20m
=
80m
Javelin
throw

24. Description of motion: Speed vs. Velocity
Speed Velocity
• A distance traveled • A displacement
over time achieved over time
• Scalar quantity • Vector quantity
– Magnitude only – Has magnitude and
direction

25. Speed vs. Velocity
10 20 30 40 50 40 30 20 10
10 20 30 40 50 40 30 20 10
A football player caught a ball at 0 yd line. He ran 52 yd and got tackled
after gaining 40 yds and moving 20 yds to the left. The play lasted 5 sec.

26. Speed = distance traveled over time
Speed = distance / Δ time
Distance = 55 yd
ΔTime = 5 sec
Speed =
distance / Δ time
= 55 yd / 5 sec
He ran 55 yds over 5 sec = 11 yd / sec

27. Velocity = displacement achieved over time
Velocity = displacement / Δ time
Displacement = 402 + 202
= 44.7 yd
ΔTime = 5 sec
20yds
Speed =
distance / Δ time
= 44.7 yd / 5 sec
40yds = 8.9 yd / sec

28. Velocity = displacement achieved over time
Alternative solution
20yd/5sec = 4yr/sec
40yd/5sec = 8yr/sec
Velocity = (8yr/sec)2 + (4yr/sec)2
= 8.9 yd / sec

29. Average vs. instantaneous speed / velocity
Average Instantaneous
• Speed or velocity • Speed or velocity at a
averaged over time specific instant
– ex. Running pace – ex. Speedometer reading
• Used when interested
“10 minute mile” in knowing
= 1mile / 10min – Max/min values
= 0.1 mile / min – Values at specific instant
= 6 mile / hour
= 6 mph

30. Instantaneous velocity
At take off, the high jumper’s vertical velocity was
9m/s and her horizontal velocity was 2m/s.
Calculate the take off velocity and angle.
? m/s
9m/s
?°
2m/s

31. Instantaneous velocity
At the instant of ball release, instantaneous velocity
of the ball was 90 mph at a direction 10° above
horizontal. What was the vertical and horizontal
velocity of the ball?
Vy
90mph
10°
Vx

32. Description of motion: Acceleration
• Rate of change in velocity
– ex. “0 to 60mph in 3 seconds”
– Vector quantity

33. Acceleration = Rate of change in velocity
Acceleration = Δ velocity / Δ time
= (Vf – Vi) / Δ time
Where:
Vf = final velocity
Vi = initial velocity

34. Acceleration = Rate of change in velocity
Sean is running a 100m dash. When the starter’s
pistol fires, he leaves the starting block and
continues speeding up until he reaches his top
speed of 11m/s 6s into the race. He holds this
speed for 2s and then gradually slows down until he
crosses the finish line at 9m/s.
What was Sean’s acceleration during:
- First 6s of the race
- From 6-8s into the race
- Last 3s

35. Acceleration = Rate of change in velocity
First 6s:
Vi = 0m/s
Vf = 11m/s
6-8s:
Vi = 11m/s
Vf = 11m/s
Last 3s:
Vi = 11m/s
Vf = 9m/s

36. (+) and (-) Acceleration
• Positive acceleration indicates that the object is
speeding up
– Acceleration
• Negative acceleration indicates that the object is
slowing down
– Deceleration

37. Acceleration due to gravity
• The rate of change in velocity caused by the
force of gravity
– 9.81m/s2 downward

38. Summary of Kinematic Descriptors
Scalar
Vector
Distance
Displacement
“Length
of
path”
“Straight
line
distance”
Speed
Velocity
“Distance
over
Fme”
“Displacement
over
Fme”
Accelera=on
“Change
in
velocity
over
Fme”
“A
rate
of
change
in
velocity”

39. CHARACTERISTICS OF
PROJECTILE MOTION

40. Projectile Motion
A rifle is shot in a perfectly horizontal plane. At
the same instant a bullet is dropped from the
same height. Which bullet hits the ground first,
the one shot from the rifle or the one dropped
next to the rifle?
Don’t
say
it,
think
about
it....

42. Projectile Motion
• Projectile is an object that has been projected into
the air or dropped and is only acted on by the
forces of gravity and air resistance
– In this unit, we consider air resistance negligible
• Examples:
– Soccer ball after impact
– Diver, long jumper, and high jumper after a take off
– Ball dropped from a top of the building

43. Projectile Motion
Apex
Trajectory (Parabolic)
Height
Distance
Release Landing

44. Effects of gravity on projectile motion
• The only force acting on projectiles is the
gravitational force (ignoring air resistance)
• Gravitational force only affects vertical velocity
– Vertical and horizontal velocity are independent!!

45. Effects of gravity on vertical velocity
• Gravity causes 9.81m/s2 acceleration in vertical
(downward) direction
– Vertical velocity of the projectile decrease by 9.81m/s
every second

46. Projectile Motion
Apex
Height (m)
0 1 2 3 4 5 6 7 8
39.2m/s
-­‐9.81
-­‐9.81
Velocity (m/s)
-­‐9.81
-­‐9.81
0
-­‐9.81
-­‐9.81
-­‐9.81
-­‐9.81
Vertical
Distance (m) velocity

47. Effects of gravity on horizontal velocity
• Gravity does not cause acceleration in horizontal
direction
– Gravity has no influence on horizontal velocity
– Horizontal velocity of the projectile does not change (=
stays constant)

48. Projectile Motion
Apex
Position (m)
0 1 2 3 4 5 6 7 8
20.0m/s
Horizontal
Velocity (m/s)
velocity
0
Time (sec)

49. Summary of the characteristics of projectile motion
• Trajectory of the center of mass (COM) of the
projectile is parabolic
– Symmetric about the apex
– Time up = time down
• Vertical velocity
– Decreases by 9.81m/s every second during up phase
– 0m/s at apex
• Horizontal velocity
– Is constant (ignoring air resistance)

50. FACTORS INFLUENCING
PROJECTILE MOTION

51. Horizontal and vertical velocity
• For the analysis of projectile motion, velocity is
often resolved into horizontal (Vx) and vertical (Vy)
component
Velocity
Vy
θ
Vx

52. Effects of projection angle on horizontal and vertical
velocity
• Increasing projection angle will:
– Decrease horizontal velocity
– Increase vertical velocity
Smaller
projec0on
angle
=
20m/s
Smaller
Vy
and
greater
Vx
20m/s
18.8m/s
10.0m/s
Greater
projec0on
angle
=
θ=70°
θ=30°
greater
Vy
and
smaller
Vx
6.8m/s
17.3m/s
Changing
projecFon
angle
will
change
the
raFo
between
horizontal
and
verFcal
velocity

53. Effects of projection angle on horizontal and vertical
velocity
• Projection speed influences the shape of
projectile’s trajectory
Steeper projection angle Milder projection angle
= taller parabola = flatter parabola

54. Effects of projection speed on horizontal and vertical
velocity
• Increasing projection speed will proportionally
increase horizontal and vertical velocity
20m/s
14.1m/s
10m/s
7.1m/s
θ=45°
θ=45°
7.1m/s
14.1m/s
Increasing
projecFon
speed
will
proporFonally
increase
both
horizontal
and
verFcal
velocity

55. Effects of projection speed on horizontal and vertical
velocity
• Projection speed influences the size of projectile’s
trajectory
Smaller projection speed Greater projection speed
= smaller parabola = greater parabola

56. Vertical and horizontal velocity
• To increase vertical velocity, you can:
1. Increase projection speed
2. Increase projection angle
• To increase horizontal velocity, you can:
1. Increase projection speed
2. Decrease projection angle

57. Variables of interest in projectile motion
1. Maximum height
• How high does the object travel?
2. Flight time
• How long does the object stay in air?
3. Flight distance
• How far does the object travel?

58. Factors Influencing Maximum Height
• Maximum height is affected by 2 factors
– Vertical velocity
– Projection height
Height (m)
0
0 1 2 3 4 5 6 7 8
Distance (m)
Increasing
verFcal
velocity
by
increasing
projec=on
velocity
will
increase
maximum
height

59. Factors Influencing Maximum Height
• Maximum height is affected by 2 factors
– Vertical velocity
– Projection height
Height (m)
0
0 1 2 3 4 5 6 7 8
Distance (m)
Increasing
verFcal
velocity
by
increasing
projec=on
angle
will
increase
maximum
height

60. Factors Influencing Maximum Height
• Maximum height is affected by 2 factors
– Vertical velocity
– Projection height
Height (m)
0
0 1 2 3 4 5 6 7 8
Distance (m)
Increasing
projecFon
height
will
increase
maximum
height

61. Variables of interest in projectile motion
1. Maximum height
• How high does the object travel?
2. Flight time
• How long does the object stay in air?
3. Flight distance
• How far does the object travel?

62. Factors Influencing Flight Time
• Flight time is affected by 2 factors
– Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing
– Vertical velocity
Shorter flight time Longer flight time
PosiFve
relaFve
height
will
NegaFve
relaFve
height
will
decrease
the
flight
Fme
increase
the
flight
Fme

63. Factors Influencing Flight Time
• Flight time is affected by 2 factors
– Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing
– Vertical velocity
RelaFve
height
only
affects
0me
down

64. Factors Influencing Flight Time
• Flight time is affected by 2 factors
– Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing
– Vertical velocity
Height (m)
0
0 1 2 3 4 5 6 7 8
Distance (sec)
Increasing
verFcal
velocity
by
increasing
projec=on
velocity
will
increase
flight
Fme

65. Factors Influencing Flight Time
• Flight time is affected by 2 factors
– Relative height of release (= final height – initial height)
• Difference in height between the time of release and landing
– Vertical velocity
Height (m)
0
0 1 2 3 4 5 6 7 8
Distance (sec)
Increasing
verFcal
velocity
by
increasing
projec=on
angle
will
increase
flight
Fme

66. Variables of interest in projectile motion
1. Maximum height
• How high does the object travel?
2. Flight time
• How long does the object stay in air?
3. Flight distance
• How far does the object travel?

67. Factors Influencing Flight Distance
• Flight distance is affected by 2 factors
– Flight time
• Given the horizontal velocity, longer the object is in air, the longer
the flight distance
– Horizontal velocity
• Given the flight time, greater the horizontal velocity, the longer
the flight distance
Distance
Speed
=
Distance
=
Speed
x
Time
Time

68. Factors Influencing Flight Distance
• Flight distance is affected by 2 factors
– Flight time
• Given the horizontal velocity, longer the object is in air, the longer
the flight distance
– Horizontal velocity
• Given the flight time, greater the horizontal velocity, the longer
the flight distance
Flight
=me
Horizontal
velocity
Increase
projecFon
speed
Increase
*
Increase
RelaFve
height
of
release
Increase
no
effect
Increase
projecFon
angle
Increase
*
Decrease
Decrease
projecFon
angle
Decrease
**
Increase
*
by
increasing
verFcal
velocity
**
by
decreasing
verFcal
velocity

69. Optimal angle of release depends on:
• Goal of the task
– High jump vs. long jump
• Projection height
– Release Ht = Landing Ht è Optimal projection θ = 45 °
• Ex: kick a ball for max horizontal displacement
– Release Ht > Landing Ht è Optimal projection θ < 45 °
• Ex: throw a ball for max horizontal displacement
– Release Ht < Landing Ht è Optimal projection θ > 45 °
• Ex: throw a ball onto elevated surface

70. Summary
Variable
Determined
by
Increased
horizontal
velocity
Increased
projecFon
speed
Decreased
projecFon
angle
Increased
verFcal
velocity
Increased
projecFon
speed
Increased
projecFon
angle
Increased
maximum
height
Increased
verFcal
velocity
Increased
projecFon
height
Increased
flight
Fme
Increased
verFcal
velocity
Increased
projecFon
height
Decreased
relaFve
height
Increased
flight
distance
Increased
horizontal
velocity
Increased
flight
Fme